The inversion of the tractrix. (English) Zbl 0685.40007
The involute of a catenary is called a tractrix. One end P of a rod PQ of fixed length L attached to a point Q which moves along the x-axis, the initial position of the rod being on the y-axis and the rod moves in such a way as to be always tangent to the path P. The problem was originally stated in physical terms: A man standing at the origin O holds a rope of length L to which a weight is attached. The man walks to the rightdragging the weight after him. When the man is at Q and the weight is at P, find the path of the weight. The elementary differential equation is \(dx/dy=-\sqrt{L^ 2-y^ 2}/y\). The solution is found to be \(x=f(y)\). This paper shows how to find y as a function of x. Involved in this inversion is an exposition of how to find the cube root of an infinite series and the use of the coefficients of reverted series.
Reviewer: B.Ross
MSC:
40E99 | Inversion theorems |
34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
References:
[1] | Ross B., Unit # 592, in: Modules and Monographs in Undergraduate Mathematics (UMAP) (1982) |
[2] | Ross, B. and Sachdeva, B. K. 1982.Table of Coefficients of Reverted Series3 |
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